Consider the following two-player game:
The game begins with k tokens placed at the number 0 on the integer number line spanning [0,n].
Each round,one player, called the chooser, selects two disjoint and nonempty sets of tokens A and B. (The sets A and B need not cover all the remaining tokens; they only need to be disjoint.) The second player, called the remover, takes all the tokens from one of the sets off the board. The tokens from the other set all move up one space on the number line from their current position.
The chooser wins if any token ever reaches n. The remover wins if the chooser finishes with one token that has not reached n.
1) If k>= 2^n , how can I find a winning strategy for the chooser?
2) If k< 2^n , how can I prove with a probabilitic method that a winning strategy for the remover exists ?
3) How can I use the method of conditional expectations to derandomize the winning strategy for the remover when k < 2^n ?